College of the Holy Cross, Fall 2021
Syllabus for Math 302 (Differential Geometry)
Professor Hwang, (rhymes with song)
Do not make travel plans that conflict with the midterm tests or your final exam. If an emergency prevents you from taking the final exam at the allotted time, speak to your Class Dean immediately to arrange for an incomplete grade, and to me to schedule a make-up exam.
The schedule below is subject to minor changes. Any substantial corrections will be announced by email and/or in class.
| Day | Date | Section | Topics |
| M | Aug 30 | Advising | |
| W | Sep 1 | Section 1.1 | Plane paths, velocity, acceleration |
| F | Sep 3 | Section 1.1 | Space paths, arc length |
| M | Sep 6 | Section 1.1 | Differential equations |
| W | Sep 8 | Section 1.2 | The Frenet frame |
| F | Sep 10 | Section 1.2 | The Frenet equations |
| M | Sep 13 | Section 1.2 | Local geometric theorems |
| W | Sep 15 | Section 1.3 | The fundamental theorem of curves |
| F | Sep 17 | Section 1.3 | Curves on the unit sphere |
| M | Sep 20 | Section 1.3 | Spherical triangles, geography |
| W | Sep 22 | Section 2.1 | Surfaces |
| F | Sep 24 | Section 2.1 | The unit normal field, regularity |
| M | Sep 27 | Section 2.1 | The first fundamental form |
| W | Sep 29 | Section 2.2 | The Gauss map and shape operator |
| F | Oct 1 | Midterm 1 | |
| M | Oct 4 | Section 2.2 | Examples and theorems, normal sections |
| W | Oct 6 | Section 2.2 | Principal and asymptotic directions |
| F | Oct 8 | Section 2.2 | Curvature, Meusnier's formula |
| M | Oct 11 | Fall Break | |
| W | Oct 13 | Fall Break | |
| F | Oct 15 | Fall Break | |
| M | Oct 18 | Section 2.2 | Surfaces of rotation |
| W | Oct 20 | Section 2.2 | Surfaces of rotation |
| F | Oct 22 | Section 2.3 | Coordinate frames, Christoffel symbols |
| M | Oct 25 | Section 2.3 | The shape operator |
| W | Oct 27 | Section 2.3 | The Codazzi and Gauss equations |
| F | Oct 29 | Section 2.3 | The Theorema Egregium, Clairaut surfaces |
| M | Nov 1 | Section 2.3 | Theorems about Gaussian curvature |
| W | Nov 3 | Section 2.3 | Liebmann's theorem |
| F | Nov 5 | Section 2.4 | Parallelism, paper surface geometry |
| M | Nov 8 | Section 2.4 | Covariant differentiation, parallel transport |
| W | Nov 10 | Section 2.4 | Geodesics |
| F | Nov 12 | Midterm 2 | |
| M | Nov 15 | Section 3.1 | Moving frames, Clairaut surfaces |
| W | Nov 17 | Section 3.1 | Geodesic curvature, holonomy |
| F | Nov 19 | Section 3.1 | The local Gauss-Bonnet theorem |
| M | Nov 22 | Section 3.1 | The Gauss-Bonnet theorem |
| W | Nov 24 | Thanksgiving | |
| F | Nov 26 | Thanksgiving | |
| M | Nov 29 | Section 3.1 | Polyhedra |
| W | Dec 1 | Section 3.2 | Hyperbolic geometry |
| F | Dec 3 | Section 3.3 | Differential forms and exterior calculus |
| M | Dec 6 | Section 3.3 | The Cartan structure equations |
| W | Dec 8 | Section 3.3 | Gaussian curvature revisited |
| F | Dec 10 | Review |